Problem: The area of the shaded region is 78 square inches. All angles are right angles and all measurements are given in inches. What is the perimeter of the non-shaded region?

[asy]size(101);
filldraw(((0,0)--(0,8)--(10,8)--(10,-2)--(6,-2)--(6,0)--cycle^^(2.5,3)--(2.5,5)--(7.5,5)--(7.5,3)--cycle),gray(.6)+fillrule(1),linewidth(1));
label("$2''$",(5.3,-1),fontsize(10pt));
label("$4''$",(8,-2.7),fontsize(10pt));
label("$2''$",(3.3,4),fontsize(10pt));
label("$10''$",(5,8.7),fontsize(10pt));
label("$10''$",(11,3),fontsize(10pt));[/asy]
Breaking the outer figure into two rectangles, we find that the total area of the shaded region plus unshaded region is $10\cdot 8 + 2\cdot 4 = 88$.  Thus the area of the non-shaded region is $88-78 = 10$ square inches.  This means that its remaining side length is 5 inches, and its perimeter is $2(2 + 5) = \boxed{14}$ inches.